Normalizer

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Normaliseur (Fr). Normalisator (Ge). Normalizzatore (It). 正規化群 (Ja).


Contents

Definition

Given a group G and one of its supergroups S, they are uniquely related to a third, intermediated group NS(G), called the normalizer of G with respect to S. NS(G) is defined as the set of all elements S ∈ S that map G onto itself by conjugation:

NS(G) := {S ∈S | S-1GS = G}

The normalizer NS(G) may coincide either with G or with S or it may be a proper intermediate group. In any case, G is a normal subgroup of its normalizer.

Euclidean vs. Affine normalizer

The normalizer of a space (or plane group) G with respect to the group E of all Euclidean mappings (motions, isometries) in E3 (or E2) is called the Euclidean normalizer of G:

NE(G) := {SE | S-1GS = G}

The Euclidean normalizers are also known as Cheshire groups.

The normalizer of a space (or plane group) G with respect to the group A of all affine mappings in E3 (or E2) is called the affine normalizer of G:

NA(G) := {SA | S-1GS = G}

"Symmetry of the symmetry pattern"

All symmetry operations of the Euclidean normalizer NE(G) map the space group onto itself. The Euclidean normaizer of a space group is therefore the group of motions that maps the pattern of symmetry elements of the space group onto itself. For this reason, it represents the symmetry of the symmetry pattern.

Euclidean normalizers of plane and space groups

For all the plane / space groups except those corresponding to a pyroelectric point group the Euclidan normalizer is also a plane / space group. Instead, plane / space groups corresponding to a pyroelectric point group have Euclidean normalizers that contain contrinuous translations in one, two or three indepenedent lattice directions: these are not plane / space groups but supergroupes of them.

Euclidean normalizers of groups with specialized metric

Plane / space groups where a specialized metric may induce a higher lattice symmetry have more than one type of Euclidean normalizer. This happens for 38 orthorhombic space groups (3 orthorhombic plane groups) as well as for the monoclinic and triclinic plane / space groups.

Example

A space group of the type Pmmm has three different Euclidean normalizers, all corresponding to basis vectors \frac{1}{2}a,\frac{1}{2}b,\frac{1}{2}c:

  • for the general case abca, NE(Pmmm) = Pmmm ;
  • if a = bc, NE(Pmmm) = P 4/mmm ;
  • if a = b = c, NE(Pmmm) = Pm(\bar 3)m.

Affine normalizers of plane and space groups

The affine normalizer NA(G) of a plane / space group G either is a true supergroup of the Euclidean normalizer of G, NE(G), or coincides with it:

NA(G) ⊇ NE(G)

Because any translation is an isometry, all translations belonging to NA(G) also belong to NE(G). Therefore, NA(G) and NE(G) necessarily have identical translation subgroups.

In contrast to the Euclidean normalizers, the affine normalizer of all plane / space groups are isomorphic groups: the type of the affine normalizer never depends on the metrical properties of the group G, as is instead the case for the Euclidean normalizers.

See also